30,042 research outputs found

### Geometry of fully coordinated, two-dimensional percolation

We study the geometry of the critical clusters in fully coordinated
percolation on the square lattice. By Monte Carlo simulations (static
exponents) and normal mode analysis (dynamic exponents), we find that this
problem is in the same universality class with ordinary percolation statically
but not so dynamically. We show that there are large differences in the number
and distribution of the interior sites between the two problems which may
account for the different dynamic nature.Comment: ReVTeX, 5 pages, 6 figure

### Brownian Occupation Measures, Compactness and Large Deviations

In proving large deviation estimates, the lower bound for open sets and upper
bound for compact sets are essentially local estimates. On the other hand, the
upper bound for closed sets is global and compactness of space or an
exponential tightness estimate is needed to establish it. In dealing with the
occupation measure L_t(A)=\frac{1}{t}\int_0^t{\1}_A(W_s) \d s of the $d$
dimensional Brownian motion, which is not positive recurrent, there is no
possibility of exponential tightness. The space of probability distributions
$\mathcal {M}_1(\R^d)$ can be compactified by replacing the usual topology of
weak convergence by the vague toplogy, where the space is treated as the dual
of continuous functions with compact support. This is essentially the one point
compactification of $\R^d$ by adding a point at $\infty$ that results in the
compactification of $\mathcal M_1(\R^d)$ by allowing some mass to escape to the
point at $\infty$. If one were to use only test functions that are continuous
and vanish at $\infty$ then the compactification results in the space of
sub-probability distributions $\mathcal {M}_{\le 1}(\R^d)$ by ignoring the mass
at $\infty$.
The main drawback of this compactification is that it ignores the underlying
translation invariance. More explicitly, we may be interested in the space of
equivalence classes of orbits $\widetilde{\mathcal M}_1=\widetilde{\mathcal
M}_1(\R^d)$ under the action of the translation group $\R^d$ on $\mathcal
M_1(\R^d)$. There are problems for which it is natural to compactify this space
of orbits. We will provide such a compactification, prove a large deviation
principle there and give an application to a relevant problem.Comment: Minor revision. To appear in the "Annals of Probability

- â€¦